Decision theory: decision matrix

The decision problem

Usually we have to choose an alternative among others, the act of deciding may carry certainty or uncertainty. Decisions are for certain when you´ve got plenty information of what you´ll do, for instance, if we want to buy clothes there´s a lot of shops, cheaper or more expensive, this decision has perfect information, so you understand the main facts that characterizes each shop (prices, quality, brands…) Otherwise, decisions could be unclear, and you´ll have to determine what you´ll choose.

A decision is a process in which you make a decision considering the consequences. These consequences are: the state of nature, which are events uncontrollable by the decider. States of nature are mutually exclusive {raining, not raining} it´s impossible to see both events to happen; and exhaustive {raining, not raining, snowing} this is out of our interest because we´re only considering “raining”. The decision (go swimming) is influenced by the states of nature coming up different kinds of outcomes depending on state of nature (go swimming or not). Another fact is the possibility of occurrence, which depends on your information or your preferences.

Decision matrix

The decision matrix is written with the two variables mentioned before, that is, the decision (di) and the state of nature (ei), where the first one are the columns, and the state of nature are rows (assuming i =1,2,3..).

For example, a decision maker has to choose to do an investment. An investment on low quality computers (d1) could cause low demand (e1) which will give us 40 (in thousands of $) revenues, comparative demand (e2) which will give us losses of 10 or high demand(e3) will provide 0 revenue. An investment on intermediate quality computers (d2) could cause a low demand (e1) giving us 0 revenue, comparative demand (e2) will provide 30 revenue and high demand (e3) 5 revenue. An investment on high quality computers (d3) could cause a low demand (e1) with 100 losses, comparative demand (e2) will give out 10 revenues, and high demand (e3) will provide 20 revenues. With this information the decision matrix A would be as follows:

We could also add to the matrix the percentage of probability of each state of nature, the vector of probabilities (P).